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Theorem ntrneifv3 40644
Description: The value of the neighbors (convergents) expressed in terms of the interior (closure) function. (Contributed by RP, 26-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
ntrnei.x (𝜑𝑋𝐵)
Assertion
Ref Expression
ntrneifv3 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠   𝑘,𝐼,𝑙,𝑚   𝑁,𝑠   𝑋,𝑙,𝑚,𝑠   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑖,𝑗,𝑠)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneifv3
StepHypRef Expression
1 dfin5 3927 . 2 (𝒫 𝐵 ∩ (𝑁𝑋)) = {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)}
2 ntrnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . 7 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneinex 40639 . . . . . 6 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
6 elmapi 8420 . . . . . 6 (𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵)
75, 6syl 17 . . . . 5 (𝜑𝑁:𝐵⟶𝒫 𝒫 𝐵)
8 ntrnei.x . . . . 5 (𝜑𝑋𝐵)
97, 8ffvelrnd 6841 . . . 4 (𝜑 → (𝑁𝑋) ∈ 𝒫 𝒫 𝐵)
109elpwid 4533 . . 3 (𝜑 → (𝑁𝑋) ⊆ 𝒫 𝐵)
11 sseqin2 4177 . . 3 ((𝑁𝑋) ⊆ 𝒫 𝐵 ↔ (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
1210, 11sylib 221 . 2 (𝜑 → (𝒫 𝐵 ∩ (𝑁𝑋)) = (𝑁𝑋))
134adantr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
148adantr 484 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑋𝐵)
15 simpr 488 . . . . 5 ((𝜑𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 40643 . . . 4 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼𝑠) ↔ 𝑠 ∈ (𝑁𝑋)))
1716bicomd 226 . . 3 ((𝜑𝑠 ∈ 𝒫 𝐵) → (𝑠 ∈ (𝑁𝑋) ↔ 𝑋 ∈ (𝐼𝑠)))
1817rabbidva 3464 . 2 (𝜑 → {𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁𝑋)} = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
191, 12, 183eqtr3a 2883 1 (𝜑 → (𝑁𝑋) = {𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼𝑠)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  cin 3918  wss 3919  𝒫 cpw 4522   class class class wbr 5053  cmpt 5133  wf 6340  cfv 6344  (class class class)co 7146  cmpo 7148  m cmap 8398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-map 8400
This theorem is referenced by: (None)
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